Lemma 9.22.3. Let L/K be a Galois extension with Galois group G. Let \Lambda be the set of finite Galois subextensions, i.e., \lambda \in \Lambda corresponds to L/L_\lambda /K with L_\lambda /K finite Galois with Galois group G_\lambda . Define a partial ordering on \Lambda by the rule \lambda \geq \lambda ' if and only if L_\lambda \supset L_{\lambda '}. Then
\Lambda is a directed partially ordered set,
L_\lambda is a system of K-extensions over \Lambda and L = \mathop{\mathrm{colim}}\nolimits L_\lambda ,
G_\lambda is an inverse system of finite groups over \Lambda , the transition maps are surjective, and
G = \mathop{\mathrm{lim}}\nolimits _{\lambda \in \Lambda } G_\lambda
as a profinite group, and
each of the projections G \to G_\lambda is continuous and surjective.
Proof.
Every subfield of L containing K is separable over K (follows immediately from the definition). Let S \subset L be a finite subset. Then K(S)/K is finite and there exists a tower L/E/K(S)/K such that E/K is finite Galois, see Lemma 9.16.5. Hence E = L_\lambda for some \lambda \in \Lambda . This certainly implies the set \Lambda is not empty. Also, given \lambda _1, \lambda _2 \in \Lambda we can write L_{\lambda _ i} = K(S_ i) for finite sets S_1, S_2 \subset L (Lemma 9.7.5). Then there exists a \lambda \in \Lambda such that K(S_1 \cup S_2) \subset L_\lambda . Hence \lambda \geq \lambda _1, \lambda _2 and \Lambda is directed (Categories, Definition 4.21.4). Finally, since every element in L is contained in L_\lambda for some \lambda \in \Lambda , it follows from the description of filtered colimits in Categories, Section 4.19 that \mathop{\mathrm{colim}}\nolimits L_\lambda = L.
If \lambda \geq \lambda ' in \Lambda , then we obtain a canonical surjective map G_\lambda \to G_{\lambda '}, \sigma \mapsto \sigma |_{L_{\lambda '}} by Lemma 9.21.8. Thus we get an inverse system of finite groups with surjective transition maps.
Recall that G = \text{Aut}(L/K). By Lemma 9.22.2 the restriction \sigma |_{L_\lambda } of a \sigma \in G to L_\lambda is an element of G_\lambda . Moreover, this procedure gives a continuous surjection G \to G_\lambda . Since the transition mappings in the inverse system of G_\lambda are given by restriction also, it is clear that we obtain a canonical continuous map
G \longrightarrow \mathop{\mathrm{lim}}\nolimits _{\lambda \in \Lambda } G_\lambda
Continuity by definition of limits in the category of topological groups; recall that these limits commute with the forgetful functor to the categories of sets and topological spaces by Topology, Lemma 5.30.3. On the other hand, since L = \mathop{\mathrm{colim}}\nolimits L_\lambda it is clear that any element of the inverse limit (viewed as a set) defines an automorphism of L. Thus the map is bijective. Since the topology on both sides is profinite, and since a bijective continuous map of profinite spaces is a homeomorphism (Topology, Lemma 5.17.8), the proof is complete.
\square
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